Fitting a Polytomous Item Response Model to Likert-Type Data

Abstract

This study examined the application of the MML-EM algorithm to the parameter estimation problems of the normal ogive and logistic polytomous response models for Likert-type items. A rating-scale model was developed based on Samejima's (1969) graded response model. The graded response model includes a separate slope parameter for each item and an item response parameter. In the rating-scale model, the item response parameter is resolved into two parameters: the item location parameter, and the category threshold parameter characterizing the boundary between response categories. For a Likert-type questionnaire, where a single scale is employed to elicit different responses to the items, this item response model is expected to be more useful for analysis because the item parameters can be estimated separately from the threshold parameters associated with the points on a single Likert scale. The advantages of this type of model are shown by analyzing simulated data and data from the General Social Surveys. Index terms: EM algorithm, General Social Surveys, graded response model, item response model, Likert scale, marginal maximum likelihood, polytomous item response model, rating-scale model. An item response model expresses a probabilistic relationship between an examinee's performance on a test item and the examinee's latent trait. Several types of item response models have been proposed. The applicability of dichotomous item response models (Birnbaum, 1968; Lord, 1980; Lord & Novick, 1968; Rasch, 1960/1980) to cognitive item response data has been extensively studied. Several polytomous item response models have been formulated based on these dichotomous models

Measurement equivalence: A comparison of methods based on confirmatory factor analysis and item response theory.

“…Several researchers (Bock, 1972;Masters, 1982;Muraki, 1990;Samejima, 1969Samejima, , 1972 have extended the dichotomous IRT models to the polytomous case. Having such models that incorporate multiple response categories is useful for analyzing Likert-type data.…”

“…One of the assumptions underlying the just-described polytomous IRT model is that the items in a scale form a unidimensional set. Additional information about these equations and polytomous IRT models can be found in Bock (1972), Masters (1982), Muraki (1990), and Samejima (1969Samejima ( , 1972. Examples of BRFs and CRFs for a five-category item are shown in Figures 1 and 2, respectively.…”

Statistical Modeling of Expert Ratings on Medical Treatment Appropriateness

“…The restricted models here are similar to those discussed by Muraki ( 1990)in the context of rating scale analysis with ordered category measures. Other models and submodels can be created with various parameter constraints.…”

Computerized Adaptive Testing with R: Recent Updates of the Package catR

“…As already mentioned, the main update of catR involves the inclusion of the most common polytomous IRT models: the graded response model (GRM; Samejima 1969Samejima , 1996, the modified graded response model (MGRM ;Muraki 1990), the partial credit model (PCM;Masters 1982), the generalized partial credit model (GPCM;Muraki 1992), the rating scale model (RSM;Andrich 1978a,b) and the nominal response model (NRM; Bock 1972). These models were integrated into the package with the following requirements and guidelines: (a) catR function names were not modified; (b) by default, all functions remain operational with dichotomous IRT models; (c) all functions support polytomous IRT models and return similar yet appropriate output.…”